A125310 Numbers n such that n = sum of deficient proper divisors of n.

6, 28, 90, 496, 8128, 33550336, 8589869056

Offset

1,1

Comments

Since any proper divisor of a perfect number is deficient, all perfect numbers are (trivially) included in the sequence.

Hence the interesting terms of the sequence are its non-perfect terms, which I call "deficiently perfect". 90 is the only such term < 10^8. Are there any more?

If a(n) were defined to be those numbers that are equal to the sum of their deficient divisors, then the sequence would begin with 1. So, up to 10^10, the only non-perfect numbers in that sequence would be 1 (a deficient number) and 90 (an abundant number). - Timothy L. Tiffin, Jan 08 2013

a(8) > 10^10. - Giovanni Resta, Jan 08 2013

These "deficiently perfect" numbers are pseudoperfect (A005835) and are a proper multiple of a nondeficient number (and hence abundant).

Links

Table of n, a(n) for n=1..7.

Index entries for sequences where any odd perfect numbers must occur

Example

90 has deficient proper divisors 1, 2, 3, 5, 9, 10, 15, 45, which sum to 90. Hence 90 is a term of the sequence.

Mathematica

sigdef[n_] := Module[{d, l, ct, i}, d = Drop[Divisors[n], -1]; l = Length[d]; ct = 0; For[i = 1, i <= l, i++, If[DivisorSigma[1, d[[i]]] < 2 d[[i]], ct = ct + d[[i]]]]; ct]; l = {}; For[i = 1, i <= 10^8, i++, If[sigdef[i] == i, l = Append[l, i]]]; l

Prog

(PARI) is(n)=sumdiv(n, d, (sigma(d, -1)<2 && d<n)*d)==n \\ Charles R Greathouse IV, Jan 17 2013

Crossrefs

Cf. A005100, A198470, A198471.

Subsequence of A005835. Fixed points of A294886. Cf. also A294900.

Sequence in context: A055711 A141255 A091321 * A336535 A342380 A138874

Adjacent sequences: A125307 A125308 A125309 * A125311 A125312 A125313

Keyword

nonn

Author

Joseph L. Pe, Mar 19 2008

Status

approved